Automatic integration using asymptotically optimal adaptive Simpson quadrature
Automatic integration using asymptotically optimal adaptive Simpson quadrature
复制标题
使用渐近最优自适应辛普森求积的自动积分
作者：
L. Plaskota
我们提出了一种新的理论方法来分析自适应求积和自适应辛普森求积，特别是这导致了一种新的自动积分算法的构建。对于给定的函数C^4f ∈C4，f^{（4）}ge 0 f（4）≥0，且可能有端点奇异性，该算法在给定的ε内渐近地产生$int _a^bf（x），{mathrm d}x$$nabf（x）dx的近似值为
八箭头0$$ε→0。此外，它在所有自适应辛普森求积中是最优的，即，需要最小数量的函数求值$$n（f，vareps）$$n（f，ε）来获得$$vareps $$ε-近似，并且运行时间与$$n（f，vareps）$$n（f，ε）成比例。
We present a novel theoretical approach to the analysis of adaptive quadratures and adaptive Simpson quadratures in particular which leads to the construction of a new algorithm for automatic integration. For a given function $$fin C^4$$f∈C4 with $$f^{(4)}ge 0$$f(4)≥0 and possible endpoint singularities the algorithm produces an approximation to $$int _a^bf(x),{mathrm d}x$$∫abf(x)dx within a given $$varepsilon $$ε asymptotically as $$varepsilon
ightarrow 0$$ε→0. Moreover, it is optimal among all adaptive Simpson quadratures, i.e., needs the minimal number $$n(f,varepsilon )$$n(f,ε) of function evaluations to obtain an $$varepsilon $$ε-approximation and runs in time proportional to $$n(f,varepsilon )$$n(f,ε).